Boundary theories of critical matchgate tensor networks

TL;DR

This paper advances the understanding of boundary theories in critical matchgate tensor networks by deriving disordered Hamiltonians that accurately represent boundary states, revealing quasiperiodic symmetries and establishing a link to conformal field theories.

Contribution

It introduces disordered local Hamiltonians that precisely model boundary states in hyperbolic tensor networks without averaging, and explores their symmetry properties and boundary-bulk correspondence.

Findings

Disordered Hamiltonians match critical Ising boundary states.

Quasiperiodic symmetries break conformal invariance in a controlled way.

Numerical evidence shows convergence to translation-invariant models at large bond dimension.

Abstract

Key aspects of the AdS/CFT correspondence can be captured in terms of tensor network models on hyperbolic lattices. For tensors fulfilling the matchgate constraint, these have previously been shown to produce disordered boundary states whose site-averaged ground state properties match the translation-invariant critical Ising model. In this work, we substantially sharpen this relationship by deriving disordered local Hamiltonians generalizing the critical Ising model whose ground and low-energy excited states are accurately represented by the matchgate ansatz without any averaging. We show that these Hamiltonians exhibit multi-scale quasiperiodic symmetries captured by an analytical toy model based on layers of the hyperbolic lattice, breaking the conformal symmetries of the critical Ising model in a controlled manner. We provide a direct identification of correlation functions of ground and low-energy excited states between the disordered and translation-invariant models and give numerical evidence that the former approaches the latter in the large bond dimension limit. This establishes tensor networks on regular hyperbolic tilings as an effective tool for the study of conformal field theories. Furthermore, our numerical probes of the bulk parameters corresponding to boundary excited states constitute a first step towards a tensor network bulk-boundary dictionary between regular hyperbolic geometries and critical boundary states.